# Extremal laws for the real Ginibre ensemble

@article{Rider2014ExtremalLF,
title={Extremal laws for the real Ginibre ensemble},
author={Brian Rider and Christopher D. Sinclair},
journal={Annals of Applied Probability},
year={2014},
volume={24},
pages={1621-1651}
}
• Published 26 September 2012
• Mathematics
• Annals of Applied Probability
The real Ginibre ensemble refers to the family of n n matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges in law to a Gumbel distribution as n!1. This fact has been known to hold in the complex and quaternion analogues of the ensemble for some time, with simpler proofs. Along the way we establish a new form for the limit law of the largest real eigenvalue.

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## References

SHOWING 1-10 OF 38 REFERENCES
Eigenvalue statistics of the real Ginibre ensemble.
• Mathematics
Physical review letters
• 2007
A computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented, relevant to May's stability analysis of biological webs.
General eigenvalue correlations for the real Ginibre ensemble
• Mathematics
• 2008
We rederive in a simplified version the Lehmann–Sommers eigenvalue distribution for the Gaussian ensemble of asymmetric real matrices, invariant under real orthogonal transformations, as a basis for
Symplectic structure of the real Ginibre ensemble
We give a simple derivation of all n-point densities for the eigenvalues of the real Ginibre ensemble with even dimension N as quaternion determinants. A very simple symplectic kernel governs both
Eigenvalue statistics of random real matrices.
• Computer Science, Mathematics
Physical review letters
• 1991
The joint probability density of eigenvalues in a Gaussian ensemble of real asymmetric matrices, which is invariant under orthogonal transformations is determined, which indicates thatrices of the type considered appear in models for neural-network dynamics and dissipative quantum dynamics.
The Ginibre Ensemble of Real Random Matrices and its Scaling Limits
• Computer Science
• 2009
We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a 2 × 2 matrix kernel
A limit theorem at the edge of a non-Hermitian random matrix ensemble
The study of the edge behaviour in the classical ensembles of Gaussian Hermitian matrices has led to the celebrated distributions of Tracy–Widom. Here we take up a similar line of inquiry in the
Averages over Ginibre's Ensemble of Random Real Matrices
A method for computing the ensemble average of multiplicative class func-tions over the Gaussian ensemble of real asymmetric matrices over GinOE, which is the space of N×Nreal matrices R.
Statistical Ensembles of Complex, Quaternion, and Real Matrices
Statistical ensembles of complex, quaternion, and real matrices with Gaussian probability distribution, are studied. We determine the over‐all eigenvalue distribution in these three cases (in the
Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble
• Mathematics
• 2008
The partly symmetric real Ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti-symmetric Gaussian random matrices. Such matrices typically have both
The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law
LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized